Section 1.4. Separable Differential Equations

Objective:

1. The definition of separable differential equation

2. Solve a separable differential equation

In this section, we learn to solve a new kind of differential equation, separable differential equation. Recall the population model, [latex]y'=ay[/latex]. This is a separable differential equation.

Definition: A first order differential equation, [latex]y'=\frac{dy}{dt}=f(y,t)[/latex], is separable if it can be presented into

\[ F(y)dy=G(t)dt\]

where [latex]F(y)[/latex] is a function of [latex]y[/latex] and [latex]G(t)[/latex] is a function of [latex]t[/latex].

Example 1: Find the general solution of the ODE: [latex]y'=\frac{y^{2}}{t}[/latex].

Exercise 1: Find the general solution of the ODE: [latex]y'=\frac{x+2}{y^{2}}[/latex] .

Example 2: Find the general solution of the ODE: [latex]y^{2}y'-\text{cos}(t)y=0[/latex].

Exercise 2: Find the general solution of the ODE: [latex]yy'+e^{t}=0[/latex].

Example 3: Solve the IVP: [latex]y^{2}y'-\text{cos}(t)y=0,y(0)=-1[/latex].

Exercise 3:  Solve the IVP: [latex]yy'+e^{t}=0,y(0)=2[/latex].

Example 4: Solve the IVP: [latex]e^{y}y'-(x+\text{sin}(x))=0,y(0)=1[/latex].

Exercise 4: Solve the IVP: [latex]x^{-2}y'-e^{x}y=0[/latex], [latex]y(0)=-2[/latex].

Group work: Solve IVP

1. [latex]y'=\frac{ty^{2}}{\sqrt{1+t^{2}}},[/latex][latex]y(0)=2[/latex].

2. [latex]y'=\frac{t^{2}}{3+2y}[/latex], [latex]y(1)=0[/latex]. 

3. [latex]ty'=\sqrt{1-y^{2}}[/latex], [latex]y(1)=1[/latex].

4. [latex]y'=\frac{x^{3}}{3y^{2}+1}[/latex], [latex]y(1)=0[/latex].

5. [latex]y'=ty+2[/latex], [latex]y(0)=3[/latex].